Question: A habitat of prairie dogs can support $m$ dogs at most. The habitat's population, $p$, grows proportionally to the product of the current population and the difference between $m$ and $p$. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{dp}{dt}=\dfrac{km}{m-p}$ (Choice B) B $\dfrac{dp}{dt}=km(m-p)$ (Choice C) C $\dfrac{dp}{dt}=\dfrac{kp}{m-p}$ (Choice D) D $\dfrac{dp}{dt}=kp(m-p)$
Solution: The population of prairie dogs is denoted by $p$. The rate of change of the population is represented by $p'(t)$, or $\dfrac{dp}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the product of $p$ and $m-p$. In conclusion, the equation that describes this relationship is $\dfrac{dp}{dt}=kp(m-p)$.